<p>Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it shows tail, then throw a dice. Find the probability of</p><p>(i) the dice shows a number greater than 4</p><p>(ii) there is at least one tail</p>
Step-by-Step Solution
Key Concept: Construct a complete sample space by tracking all possible outcomes: either the coin shows H (then toss again) or shows T (then roll dice). Use the law of total probability by conditioning on the first coin toss outcome.
<p><strong>Understanding the Experiment:</strong> First toss a coin. If H → toss coin again. If T → throw a dice.</p><p><strong>Complete Sample Space:</strong></p><p>• HH (prob = 1/2 × 1/2 = 1/4)</p><p>• HT (prob = 1/2 × 1/2 = 1/4)</p><p>• T1, T2, T3, T4, T5, T6 (each with prob = 1/2 × 1/6 = 1/12)</p><p><strong>Part (i): Probability that dice shows a number greater than 4</strong></p><p>The dice is thrown only when first toss is T. Numbers greater than 4 are {5, 6}.</p><p>P(dice > 4) = P(T5) + P(T6) = 1/12 + 1/12 = <strong>1/6</strong></p><p><strong>Part (ii): Probability of at least one tail</strong></p><p>At least one tail means: T occurs at first toss OR T occurs at second toss.</p><p>Outcomes with at least one tail: {HT, T1, T2, T3, T4, T5, T6}</p><p>P(at least one tail) = 1/4 + (1/12 + 1/12 + 1/12 + 1/12 + 1/12 + 1/12)</p><p>= 1/4 + 6/12 = 1/4 + 1/2 = <strong>3/4</strong></p><p><strong>Alternatively:</strong> P(at least one tail) = 1 - P(no tail) = 1 - P(HH) = 1 - 1/4 = 3/4 ✓</p>
Correct Answer: (i) 1/6, (ii) 3/4